10.19.2 Tensor Non Tensorial Coordinate Dependence

Tensor Non Tensorial Coordinate Dependence is the property, characteristic of quantities that fail to obey the tensorial transformation rule, whereby the individual numerical value of a single component depends not only on the underlying geometric or physical situation but also on the arbitrary choice of coordinate system used to describe it, so that the same quantity can be made to vanish or take any prescribed value at a chosen point simply by selecting a suitable coordinate system.

The Core Phenomenon

Value Tied to Coordinate Choice

A quantity exhibiting non-tensorial coordinate dependence does not possess a single well-defined numerical value at a point independent of coordinates; rather, its value is genuinely different depending on which coordinate system is used to compute it, a behavior made explicit by the presence of the extra, non-cancelling term in its transformation law:

confirming that the barred and unbarred components genuinely disagree with what the ordinary tensorial rule alone would predict.

Vanishing at a Chosen Point

Because the extra term in the transformation law depends on the second derivatives of the transformation map, and these can be freely prescribed at a single point by choosing an appropriately curved coordinate map, a non-tensorial quantity's value at that one point can always be set to zero by a suitable local choice of coordinates, a freedom that has no counterpart for a genuine tensor, whose vanishing or non-vanishing at a point is a coordinate-independent fact.

Contrast With Tensorial Coordinate Independence

Tensors Fix an Invariant Fact

For a genuine tensor, the statement "this tensor vanishes at this point" is a coordinate-independent fact, true in every coordinate system if true in one, since the tensorial rule is homogeneous and linear in the tensor's own components, so a value of zero maps to zero under any change of basis.

Non-Tensors Fix No Such Fact

For a non-tensorial quantity, no equivalent coordinate-independent statement about vanishing exists in general, since the transformation law's extra term can convert a zero value in one coordinate system into a non-zero value in another, and conversely can convert a non-zero value into zero, making "vanishing at a point" a coordinate-dependent rather than a geometric statement for such a quantity.

Practical Consequence: Locally Flattening Coordinates

Choosing Coordinates to Simplify a Point

The freedom to eliminate a non-tensorial quantity at a single point via coordinate choice is exploited deliberately in constructing special coordinate systems, such as one in which the connection coefficients vanish at a chosen point, simplifying local computations there, even though the connection coefficients generally remain non-zero at every other point of the same coordinate system.

Limits of the Simplification

This simplification is inherently local: no coordinate choice can generally make a non-tensorial quantity such as the connection coefficients vanish over an entire extended region unless the underlying space has a special property, such as flatness, that makes such a global simplification possible; away from that special case, the coordinate dependence reasserts itself as soon as one moves away from the single chosen point.

Diagram of Coordinate-Dependent Value

Same Point, Different Numbers

Testing for Coordinate Dependence

Comparing Values Across Two Arbitrary Charts

A direct way to expose non-tensorial coordinate dependence in an unfamiliar quantity is to compute its value at a fixed point in two genuinely different coordinate systems and check whether the results agree once properly related by the ordinary tensorial transformation formula; a persistent mismatch, traceable to an extra term involving second derivatives, confirms the presence of non-tensorial coordinate dependence rather than a computational error.

Distinguishing From Rounding or Computational Error

Because non-tensorial coordinate dependence produces a systematic, formula-predictable discrepancy rather than a small numerical error, it can be distinguished from an ordinary arithmetic mistake by checking whether the observed discrepancy matches exactly the extra term predicted by the known transformation law of the quantity in question, rather than appearing as an arbitrary or inconsistent difference.